Why is the product of negative numbers positive: Proof with examples
The product of negative numbers are positive because multiplying a negative number by another negative number, the resultant operation is positive in nature. Algebra is the branch of mathematics dealing with arithmetic operations and their associated symbols. The symbols are termed as variables that may take different values when subjected to different constraints.
The variables are mostly denoted such as x, y, z, p, or q, which can be manipulated through different arithmetic operations of addition, subtraction, multiplication, and division, in order to compute the values.
Why is the product of negative numbers positive?
Answer:
Upon multiplication of a negative number by another negative number, the resultant operation is positive in nature.
To Prove: The product of two negative numbers or terms is positive:
(−a)(−b) = ab
where, a and b can be:
- Numbers (i.e. a = 5, b = 1/2)
- Constants
- Variables
- Expressions [i.e. a = (y2 + 6), b = (h − w + z)]
Proof
To prove (−a)(−b) = ab, we can consider the equation:
x = ab + (−a)(b) + (−a)(−b)
It can be easily shown that x = ab and x = (−a)(−b).
Factor out −a
First, factoring out −a from the expression (−a)(b) + (−a)(−b):
x = ab +(−a)(b) + (−a)(−b)
Thus, we obtain,
x = ab + (−a)[b + (−b)]
Since, b + (−b) = 0
x = ab + (−a)(0)
Thus,
x = ab
Factor out b
Factoring out b from the expression ab + (−a)(b):
x = ab + (−a)(b) + (−a)(−b)
x = b[a + (−a)] + (−a)(−b)
x = b(0) + (−a)(−b)
Therefore,
x = (−a)(−b)
Result
Since x = ab and x = (−a)(−b):
(−a)(−b) = ab
This can be extended to any even amount of negative numbers by factoring out in steps:
(−a)(−b)(−c)(−d) = ab(−c)(−d) = abcd
Summary
The method easily proves (−a)(−b) = ab.
The fact that the product of two negative numbers, terms, or expressions is positive can be extended to any even number of negative items.
Sample Questions on product of negative numbers
Question 1. Find the product of -3a × -20b
Solution:
Here we have to find the product of -3a × -20b
As we know that when we multiply two negative numbers the answer will be positive
Now multiplying
-3a × -20b = +60ab
Therefore, the solution is positive.
Question 2. Evaluate 2ab + (−a)(5b) + (−2a)(−3b)?
Solution:
Here we have to find the product of
= 2ab + (−a)(5b) + (−2a)(−3b)
First solving the brackets
= 2ab + (-a × 5b) + (-2a × -3b)
= 2ab + (-5ab) + 6ab
= 2ab – 5ab + 6ab
= 8ab – 5ab
= 3ab
Question 3. Evaluate 10ab + (−21a)[5b + (−10b)]?
Solution:
Here we have to find the product of 10ab + (−21a)[5b + (−10b)]
First solving the brackets
= 10ab + (−21a)× [5b −10b]
= 10ab – 21a × (-5b)
= 10ab + 105ab
= 115ab
Question 4. Find the product of {-(4x + 5x) × (12x – 16x)}?
Solution:
Here we have to find the product of {-(4x + 5x) × (12x – 16x)}
First solving the brackets
= {(-4x – 5x) × (12x – 16x)}
= (-9x) × (-4x)
= 36x2
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